Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

TL;DR

This paper introduces a new invariant called the generating function barcode for Hamiltonian diffeomorphisms in symplectic topology and provides an algorithm to approximate it using Morse theory.

Contribution

It defines the generating function barcode for compactly supported Hamiltonian diffeomorphisms and develops a finite algorithm to approximate this invariant.

Findings

The generating function barcode can be approximated algorithmically.

Morse theory applied to generating functions provides a practical computational approach.

The method advances numerical tools in symplectic topology.

Abstract

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of $ \mathbb{R}^{2n} $ by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it.